Topics
8. Statistics
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United States of America
Grade 6

8.06 Mean absolute deviation (MAD)

Lesson
Practice

Introduction

Measures of variability tell us how far the values in a data set are spread out. We've already looked at one measure of spread, the  range  , which is the difference between the greatest and least value in a data set. Now we are going to learn about a new measure of spread called the Mean absolute deviation (MAD).

Mean absolute deviation (MAD)

The mean absolute deviation (MAD) of a set of data is the average distance between each data value and the mean. Why would we want to look at this average distance? Well, by calculating the average distance of each data value from the mean, we can see if the data values are close together or far apart.

If the MAD value is small, that tells us the average distance of the values from the mean is small, therefore the data values are closer together. If the MAD value is large, then we know the distance of the values from the mean is large, therefore the data values are more spread apart.

To calculate the mean absolute deviation of a set of data:

1. Calculate the mean.

2. Find the absolute value of the differences between each value in the set and the mean.

3. Find the average of those values.

Examples

Example 1

Find the mean absolute deviation of the following data set:2,\,8,\,6,\,3,\,10,\,15,\,6,\,6

a

First, find the mean.

Worked Solution
Create a strategy

Find the average of the numbers by using the formula: \text{mean}=\dfrac{\text{sum of values}}{\text{number of values}}

Apply the idea
\displaystyle \text{mean}\displaystyle =\displaystyle \dfrac{2+8+6+3+10+15+6+6}{8}Substitute all the values
\displaystyle =\displaystyle \dfrac{56}{8}Evaluate the addition
\displaystyle =\displaystyle 7Evaluate the division
b

Complete the table of values, finding the distance of each value from the mean.

\text{Value}\text{Distance from } 7
2
8
6
3
10
15
6
6
Worked Solution
Create a strategy

Subtract each value in the table from the mean.

Apply the idea

Subtract each value and take the absolute value:\begin{aligned}|2-7|&=&5\\ |8-7|&=&1\\|6-7|&=&1\\|3-7|&=&4\\|10-7|&=&3\\|15-7|&=&8 \end{aligned}

\text{Value}\text{Distance from } 7
25
81
61
34
103
158
61
61
c

Using your values from the table above, calculate the mean of the differences.

Worked Solution
Create a strategy

Find the average of the distances from the table.

Apply the idea
\displaystyle \text{Mean of differences}\displaystyle =\displaystyle \dfrac{5+1+1+4+3+8+1+1}{8}Substitute all differences from the table
\displaystyle =\displaystyle \dfrac{24}{8}Evaluate the addition
\displaystyle =\displaystyle 3Evaluate the division

Example 2

Which of the following is true concerning the mean absolute deviation of a set of data?

A
It describes the average distance between each data value and the mean.
B
It describes the variation of the data values around the median.
C
It describes the absolute value of the mean.
D
It describes the variation of the data values around the mode.
Worked Solution
Create a strategy

Choose the option that best describes the mean absolute deviation.

Apply the idea

The correct option is A, because mean absolute deviation describe the average distance between each data value and the mean.

Idea summary

The mean absolute deviation (MAD) of a set of data is the average distance between each data value and the mean.

To calculate the mean absolute deviation of a set of data:

1. Calculate the mean.

2. Find the absolute value of the differences between each value in the set and the mean.

3. Find the average of those values.

If the MAD value is small, our data values are closer together. If the MAD value is large, then our data values are more spread apart.

Outcomes

6.SP.A.2

Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

6.SP.A.3

Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number

6.SP.B.5

Summarize numerical data sets in relation to their context, such as by:

6.SP.B.5.C

Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data was gathered.

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